Motivated by questions about the behavior of solutions of elliptic and parabolic equations with low regularity drift terms, Seregin, Silvestre, Šverák, and Zlatoš ([1]}) investigated equations such as \(-\Delta u+\mathbf {c}\cdot \nabla u=0\)  and \(\partial _tu+\mathbf {c}\cdot \nabla u-\Delta u=0\) , where \(c\)  is a divergence-free vector field in \(\mathbb {R}^n\) . They discovered that the divergence-free condition can be utilized to relax the regularity assumptions on \(\mathbf {c}\)  under which one can obtain the Harnack inequality and other regularity results for solutions. It turns out that the interior regularity theory of De Giorgi, Nash, and Moser can be carried over to elliptic equations with \(\mathbf {c}\in \operatorname{BMO}^{-1}\) , and to parabolic equations with \(\mathbf {c}\in L^{\infty }(\operatorname{BMO}^{-1})\) . Generalizing to elliptic or parabolic equations in divergence form, this condition is equivalent to assuming that the coefficient matrix \(A\)  of the operator \(L=-\operatorname{div}(A\nabla )\)  can be decomposed into an \(L^{\infty }\)  elliptic symmetric part and an unbounded anti-symmetric part in a certain function space. To be precise, the anti-symmetric part should belong to the John-Nirenberg space \(\operatorname{BMO}\)  (bounded mean oscillation) in the elliptic case, and to \(L^{\infty }(\operatorname{BMO})\)  in the parabolic case.
The space \(\operatorname{BMO}\)  plays a key role in two ways. First, this space has the right scaling properties which arise naturally in the iterative arguments of De Giorgi-Nash-Moser. Secondly, the \(\operatorname{BMO}\)  condition on the anti-symmetric part of the matrix enables one to properly define weak solutions. This latter fact follows essentially from the div-curl lemma appearing in the theory of compensated compactness ([2]}), and the details can be found in [1]} or [4]}.
